Can a bad thermostat cause you to rapidly loose antifreeze?

Yes, because as the water heats it expands. The water system will then spill over to the expansion tank. The expansion tank is only big enough to store expanded water at the thermostat temperature. When your thermostat doesn't open the water will over heat/ over expand. You will see brownish burnt water marks all over the expansion tank near the overflow hole. Best off just throw away your thermostat you don't really need it.